Exercise 573Logic, sets and proofs › Sets and mapsHigh schoolAdded 3 Nov 2026

Cartesian product

#Cartesian product#ordered pair#cardinality

Statement

Let $A=\{1,2\}$ and $B=\{a,b,c\}$ . Compute $A\times B$ and $B\times A$ . Compare their cardinalities.

Hint

$A\times B=\{(x,y):x\in A,\ y\in B\}$ .

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$A\times B$ . $\{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\}$ .

$B\times A$ . $\{(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)\}$ .

Cardinalities. $|A\times B|=2\cdot3=6$ , $|B\times A|=3\cdot2=6$ . Equal, but the sets differ (different elements).

Conclusion. $A\times B\neq B\times A$ in general, though $|A\times B|=|B\times A|$ .